189
Inat. J. Math Mh. Si.
(1980) 189-192
Vol. 3 No.
A COVERING THEOREM FOR ODD TYPICALLY-REAL FUNCTIONS
E.P. MERKES
Department of Mathematical Sciences
University of Cincinnati
45221 U.S.A.
Cincinnati, Ohio
(Received July 20, 1979)
ABSTRACT.
if
An analytic function f(z)
Im f(z)Im
z > 0.
z
+ aoz 2 +...
in
Izl
< I is typically-real
The largest domain G in which each odd typically-real
function is univalent (one-to-one) and the domain
(G) for all odd typically
real functions f are obtained.
KEY WORDS AND PHRASES.
Typicy-real funtns, dmain
of
threms
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES.
1.
univalence, coving
30C25.
INTRODUCTION.
An analytic function f(z)
z
+
amZ2 +...
in the unit disk E
(Izl
< i) is in
the class T of typically-real functions if and only if there exists a nondecreas-
ing function y on [0,7] such that y()
f(z)
I
0 1
i, y(0)
0, and
gay(t)
2z cos t
+
z
The function y when normalized on (0,7) by y(t)
uniquely determined by f.
(y(t+) + y(t-))/2 is
E.P. MARKES
190
The domain of unlvalence of the class T is known [23 to be
(1.2)
Brannon and Krwan [3] proved that the largest domain contained in f(G) for
[w
every function in T is
<
1/4.
In this paper we obtain the corresponding results for the class T O of odd
Recently Goodman [4] determined the largest domain
typlcally-real functions.
The analog of this result for the
that is contained in f(E) for every f e T.
class T
2.
O
is an open problem.
The domain of unlvalence of T
THEOREM 2.1.
PROOF.
O
the domain G of (1.2).
O is
is univalent in G. The theorem is estab-
The domain of unlvalence for T
Since T
O
c T, each f e T
O
lished, therefore, if we can show that there is a function f e T O that is not
Let f(z)
univalent in any domain D that properly contains G.
2
z(l + z )/(I
z
2 2
i
z/(l
since T is a linear class.
maps G onto
II
< i.
z)
2
i
+
z/(l
-
+ z) 2
This function is clearly in T O
The function
2z
l+z
(2.1)
2
By the change of variables (2.1), the function f has the
form
f(z)
=Iz/(l
1
: 1 (1
Since
/(I-
2)
2z
+
:) +
z
1
2)
+
;I(1
1/2z/(l + 2z + z 2
+ )
1
/(I
2
)
is not univalent in any domain that properly contains
II
< i,
we conclude that f is not univalent in any domain that properly contains G.
3.
A covering theorem for T O
The largest domain U contained in f(G) for every f e T O is the
18
pe
domain that includes the origin, is bounded in the right half-plane by w
THEOREM 3.1.
191
COVERING THEOREM FOR REAL FUNCTIONS
where
0
f
o
(cs 0)/2
<
1el
<
T/4,
and is symmetric relative to the imaginary axis.
By, (i. I) we have
PROOF.
_f(_z)
I
0 1
+ 2z
+
cos t
z
2
zd[l
0
>(
2z cos
1
T)]
T
+ z2
1/2.
In particular, y(/2)
[0,].
For f e T O
/2
f(z)
I
I
zdT(t)
I
0
2z cos t
+ z2
+I
2
t) for
therefore,
i- 2z cos t
/2
y(
I
-f(-z), then by the uniqueness of y we have y(t)
If f(z)
t e
I
zdy(t)
+
z
;
0
i
0
2z
zd(t).
cos t +
z
2
+
zdy( -t)_
2
cos (- t) + z
I2z
/2
/2
z
I-
0
2z cos t
+
z
2
z
2z cos t
+
i
+
+
z
2.]dy (t)
By the change of variables (2.1), we obtain
/2
f(z)
I
i
0
Let z e G.
II
i.
2
COS
2
By (2.1) we have that the corresponding
For fixed
e
i8
-
,
0 <
< l, the point le
< 8
io
(3.z)
0
t
,
+ (1
is on the unit circle
the function 2f(z) is by (3.1) in the
closed convex hull H of the circular arc w(s)
each
1/2, y(0)
y(/2)
dT(t)
e
iO
/ (1
se
2iO)
s
[0,1].
%)i/sin O is on the linear portion of H.
Let D(%) denote the square of the distance from such a point to the origin.
O
# 0,,
we have
For
If
E.P. MERKES
192
D(-I)
xe ie
/
1
(1
X)i/sin
el 2
i- 2
This function of I has a minimum at
Isin el
<
than [D(I
8 e
/2.
1
+
(1-X2)/(4 sln2e)
sin28
and 0 < A < i provided
Thus, the distance from any point of H to the origin is not less
sln28)] 1/2
Icos 81
when 0 <
181
<_ /4
or
3/4 <_
181
< w.
For other
(-,] the distance from any point of H to the origin is
minfl,1/(21sin el)}
1/(21sin el)
for /4 <
iel
<
3/4 and 1 for 8
Since the convex hull H contains for each z e aG the values of
0 or
.
2f(z) for all
f e T and since every point of H is the value of 2f(z) for some f e T O
O
conclude that U is the exact domain covered by all f e T O when z
we
G.
REFERENCES
i.
Robertson, M. S., On the coefficients of a typically-real function,
Bull. Amer. Math. Soc., 41 (1935), 565-572.
2.
Merkes, E. P., On typically-real functions in a cut plane, Proc. Amer.
Math. Soc., i0 (1959), 863-868.
3.
Brannan, D. A. and Kirwan, W. E., A covering theorem for typically-real
functions, Glasgow Math. J., i0 (1969), 153-155.
4.
Goodman, A. W., The domain covered by a typically-real function, Proc. Amer.
Math. Soc.., 64 (1977), 233-237.